L(s) = 1 | + (−1.48 + 1.67i)5-s + i·7-s + 0.387·11-s − 2.96i·13-s + 3.35i·17-s − 2.96·19-s − 0.962i·23-s + (−0.612 − 4.96i)25-s + 1.22·29-s − 2.96·31-s + (−1.67 − 1.48i)35-s − 5.92i·37-s − 1.03·41-s − 10.7i·43-s − 3.22i·47-s + ⋯ |
L(s) = 1 | + (−0.662 + 0.749i)5-s + 0.377i·7-s + 0.116·11-s − 0.821i·13-s + 0.812i·17-s − 0.679·19-s − 0.200i·23-s + (−0.122 − 0.992i)25-s + 0.227·29-s − 0.532·31-s + (−0.283 − 0.250i)35-s − 0.974i·37-s − 0.162·41-s − 1.63i·43-s − 0.470i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.153934384\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.153934384\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.48 - 1.67i)T \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 - 0.387T + 11T^{2} \) |
| 13 | \( 1 + 2.96iT - 13T^{2} \) |
| 17 | \( 1 - 3.35iT - 17T^{2} \) |
| 19 | \( 1 + 2.96T + 19T^{2} \) |
| 23 | \( 1 + 0.962iT - 23T^{2} \) |
| 29 | \( 1 - 1.22T + 29T^{2} \) |
| 31 | \( 1 + 2.96T + 31T^{2} \) |
| 37 | \( 1 + 5.92iT - 37T^{2} \) |
| 41 | \( 1 + 1.03T + 41T^{2} \) |
| 43 | \( 1 + 10.7iT - 43T^{2} \) |
| 47 | \( 1 + 3.22iT - 47T^{2} \) |
| 53 | \( 1 - 5.66iT - 53T^{2} \) |
| 59 | \( 1 + 3.22T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 5.53T + 71T^{2} \) |
| 73 | \( 1 - 6.18iT - 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 - 3.22iT - 83T^{2} \) |
| 89 | \( 1 - 3.73T + 89T^{2} \) |
| 97 | \( 1 - 7.73iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.304676797905347284150352224852, −7.35683412225918839775100142685, −6.83442638872407963500223672436, −5.96871289247508842849324494015, −5.37774239035769092835894765483, −4.23745412274369756692447816449, −3.67286586839929888059526103492, −2.78598502913500036042658295919, −1.94245463844161543277341908029, −0.39496115836632745959942262345,
0.859945658357297931804624746460, 1.88071727369130462449103292935, 3.08336073777974469444590746386, 3.96257363586955074180395428124, 4.60324614554178138910536370500, 5.19283716061223600030351701222, 6.27879721488225893325101772753, 6.92269990339413392860888827580, 7.66701419936897244927441736016, 8.289379093269366632569198674236